Introduction to Climate Modelling

Thomas Stocker

Introduction to Climate Modelling doesn't attempt a systematic
introduction to climate modelling, but focuses on the fundamental dynamics
of the atmospheric and oceanic circulations, along with approaches to
their mathematical formulation and numerical treatment. Some radiation
balance models are covered, but there's nothing on atmospheric or oceanic
chemistry, vegetation, ice sheets, or other components of the broader
earth system.

With that limitation, a broad range of models are treated, chosen both
for their scientific significance and broader interest and to illustrate
key concepts and mathematical tools. Though no implementation details are
considered, the treatment is moderately technical and assumes familiarity
with simple vector and multivariable calculus, differential equations,
and fluid mechanics. (Introduction to Climate Modelling is used as a
text for a one semester graduate course, aimed at students who may have
only a basic knowledge of climate science but who have a background in
physics, or perhaps in engineering or applied mathematics.)

A rapid introduction covers the basics of the climate system, the
purpose and limitations of modelling and something of its history, some
examples of current climate models, and the hierarchy of models (where
"more complex" is by no means "better").

A basic zero-dimensional radiative balance model produces a simple
analytic solution, which then illustrates the use of finite difference
methods in finding "the numerical solution of an ordinary differential
equation of first order". Context for this model is given in an overview
of the climate sensitivity and the major feedbacks to greenhouse forcing.

Turning to energy and matter transport, Stocker looks at equations
for diffusion, advection, and advection-diffusion; numerical solutions
to a simplified advection equation are then derived. This introduces
numerical stability and the Courant–Friedrichs–Lewy criterion, as
well as different discretization schemes — "Euler forward in time,
centered in space" and so forth — and touches on the problems with
non-physical "numerical diffusion" produced by approximations.

Turning to energy transport on a larger scale, Stocker presents some
simple meridional energy balance models. Atmospheric heat transport can
be partitioned into a mean meridional flow and fluxes due to stationary
and transient eddies; the ocean heat transport can be partitioned into
ocean gyres, meridional overturning circulation, Ekman circulation,
and eddy diffusivity. In both cases, "sub-scale transports need to be
parametrised due to the limitations imposed by the grid resolution".

With the large scale ocean circulation, Stocker considers the equations
of motion and continuity, the special case of shallow water equations,
and the Stommel model for flows driven by the wind. Some of the
mathematical methods introduced include the use of different reference
frames, initial and boundary value conditions, iterative methods, and
grids; spectral models are briefly touched on.

A simplified approach to the general circulation of the atmosphere shows
how meridional flows can involve thermally direct (Hadley) and indirect
(Ferrel) cells. The Lorenz-Saltzman model is presented as an example
of a chaotic system that can generate spontaneous abrupt changes or
self-sustained oscillations.

A final chapter looks at the possibilities of multiple equilibria — with
"tipping points" between them — taking as an example the abrupt changes
found in polar ice cores and their possible explanation by a bipolar
seesaw in the Atlantic circulation, and applying coupled models to the
latter's future under greenhouse warming. The possibility of multiple
equilibria in a simple atmospheric energy balance model ("Snowball Earth"
scenarios) is also touched on.

Introduction to Climate Modelling could be used to lure students from
other disciplines to research on circulation models, but is perhaps
most valuable for offering a plunge — more than a shallow dip — into
the topic for the much broader range of scientists who will use climate
models in their work and could benefit from an understanding of their
internal complexities. It also makes a nice approach for anyone else
with the necessary background in physics and mathematics who wants a
better understanding of an area of science which has taken on a role in
important public policy decisions.